An optimal power control algorithm based on convex optimization is proposed for base stations in long term evolution networks. An objective function was formulated to maximize the proportional fairness of the networks. The optimal value of the objective function was obtained using convex optimization and distributed methods based on the path loss model between the base station and users. Field tests on live networks were conducted to evaluate the performance of the proposed algorithm. The experimental results verified that, in a multi-cell multi-user scenario, the proposed algorithm increases system throughputs, proportional fairness, and energy efficiency by 9, 1.31 and 20.2 %, respectively, compared to the conventional fixed power allocation method.